In
this section a technique is described for acquisition of pointlike
objects using a very low cost spectrograph. The application field
would be to study low resolution supernovae spectra and novae spectra,
but many other object types would be within reach of the rudimentary
spectrograph proposed here, such as comets, planetary nebulae or
quasars.

Click
here for more informations about the use of this method with
a digital SLR (Star Analyser grating)

In order to achieve this goals, the
diffraction grating is located directly in the telescope's converging
beam, close to the CCD's sensitive surface. With regard to optical
aberrations, this is not the best design to be considered (further
on we will see some improvements). But such an arrangement provides
for an extremely simple and efficient instrument, often called a
"field spectrograph" or "imager spectrometer",
that can be used on weak objects.

The resolution power parameter (R)
is the ratio between wavelength l and the spectral separation power
at this wavelength, Dl :

Our modest spectrograph will allow
us to reach R=50 to 100, while some professional slit spectrographs
have a resolution in excess of 30,000. However, with this modest
resolution, our spectrograph will deliver more usable information
than images made using a set of BVRI filters (10 times as many spectral
channels since a filter in a BVRI system typically, has a Full Width
at Half Maximum - FWHM of 1000A).

Figure below show the grating's position
in the optical beam. Depending on wavelength, polychromatic light
from a field star is spread differently when going through the array.
Hence a spectrum appears in the CCD's plane. Each field object has
its own spectrum: stars, galaxies...

Disposition of the grating in the telescope's
converging beam.

The deviation angle for a given wavelength
light ray is given by:

In this formula a is
the angle of the incident ray respective to a line normal
to the surface of the grating, b is the angle of emergence after diffraction,
m is the number of ruled grooves per millimiter (in
general, m is comprised between 50 and 1200 grooves/mm),
k is an integer number for the spectrum order and l is
the wavelength in millimeter.

If on average the incident beam is
normal to the surface (as is the case in the preceding figure),
then a=0, and we obtain a simplified version of the previous
formula:

Let us assume that the grating has
100 grooves per mm and that we are studying the path of a ray at
a wavelength of 0,65 micron or 0,65.10-3 mm. For k=0, obviously b is 0 whatever the
observed wavelength. This means that through the grating we observe
a direct, polychromatic image of the object, with no deviation.

For k=1, angle b is given by

For k=-1, we find a deviation angle
of –3,7°. This means that on either side of the 0 order we find
two identical spectra.

It can easily be verified that for
larger values of k other spectra appear symmetrically, farther
from the 0 order image. In good quality gratings, grooves are given
a shape that concentrate most of the light in a given order different
from 0 (this is the grating's blaze angle).

The diffraction of an HeNe laser's beam
when passing through a grating. The screen shows as many monochromatic
images of the beam as there are orders. One of the images is much
brighter than its neighbors, indicating approximatively the blaze
angle (note that this is not the zero order).

On
either side of Vega's zero order image (saturated here, hence the
vertical blooming streak), are spectra of orders 1 and –1.
Further on higher orders can be found, but they are too weak to
give a clear visible signal in this CCD image reproduction. This
observation was obtained with a #40 Cokin grating, and an Audine camera with a KAF-0400.